Abstract

We discuss the uniqueness of quantum states compatible with given
measurement results for a set of observables. For a given pure
state, we consider two different types of uniqueness: (1) no other
pure state is compatible with the same measurement results and (2)
no other state, pure or mixed, is compatible with the same
measurement results. For case (1), it was known that for a
d-dimensional Hilbert space, there exists a set of 4d−5
observables that uniquely determines any pure state. We show that
for case (2), 5d−7 observables suffice to uniquely determine any
pure state. Thus there is a gap between the results for (1) and (2),
and we give some examples to illustrate this. Unique determination
of a pure state by its reduced density matrices (RDMs), a special
case of determination by observables, is also discussed. We improve
the best known bound on local dimensions in which almost all pure
states are uniquely determined by their RDMs for case (2). We
further discuss circumstances where (1) can imply (2). We use
convexity of the numerical range of operators to show that when only
two observables are measured, (1) always implies (2). More
generally, if there is a compact group of symmetries of the state
space which has the span of the observables measured as the set of
fixed points, then (1) implies (2). We analyze the possible
dimensions for the span of such observables. Our results extend
naturally to the case of low rank quantum states.

pacs:

In a d-dimensional Hilbert space Hd, the description of
any quantum state ρ generated by a source can be obtained by
quantum tomography. For any density matrix ρ, which is Hermitian
and has trace 1, d2−1 independent measurements are sufficient and
necessary to uniquely specify ρ. When ρ=|ψ⟩⟨ψ|
is a pure state, one may not need as many measurements to uniquely
determine |ψ⟩. As we will see later, however, exactly what is
meant by “uniquely” in this context needs to be specified.

Consider a set of m linearly independent observables

A=(A1,A2,…,Am)

(1)

where each Ai is Hermitian. Measurements on state ρ with
respect to these observables give the following average values

A(ρ):=(trρA1,trρA2,…,trρAm)∈Rm.

(2)

We denote the set of these A(ρ) for all states ρ as

Cm(A):={A(ρ):ρ acts on Hd}.

(3)

For a pure state |ψ⟩, these values are given by

A(|ψ⟩):=(⟨ψ|A1|ψ⟩,⟨ψ|A2|ψ⟩,…,⟨ψ|Am|ψ⟩),

(4)

and we denote the set of these values for all pure states |ψ⟩
as the joint numerical range

Wm(A):={A(|ψ⟩):|ψ⟩∈Hd}.

(5)

In this work we consider two different kinds of “unique
determinedness” for |ψ⟩:

We say |ψ⟩ is uniquely determined among pure
states (UDP) by measuring A if there does not exist
any other pure state which has the same measurement results as those
of |ψ⟩ when measuring A.

We say |ψ⟩ is uniquely determined among all
states (UDA) by measuring A if there does not exist
any other state, pure or mixed, which has the same measurement
results as those of |ψ⟩ when measuring A.

It is known that there exists a family of 4d−5 observables such that
any pure state is UDP, in contrast to the d2−1 observables in the
general case of quantum tomography Heinosaari et al. (2011). The physical meaning
for this case is clear: it is useful for the purpose of quantum
tomography to have the prior knowledge that the state to be
reconstructed is pure or nearly pure. Many other techniques for pure
state tomography have been developed, and experiments have been
performed to demonstrate the reduction of the number of measurements
needed Weigert (1992); Amiet and Weigert (1999); Finkelstein (2004); Flammia et al. (2005); Gross et al. (2010); Cramer et al. (2010); Liu et al. (2012).

When the state is UDP, to make the tomography meaningful, one needs to
make sure that the state is indeed pure. This is not in general
practical, but one can readily generalize the above mentioned UDP
results to low rank states, where the physical constraints (e.g., low
temperature, locality of interaction) may ensure that the actual
physical state (which ideally supposed to be pure) is indeed low rank.
If the state is UDA, however, in terms of tomography one do not need
to bother with these physical assumptions, because in the event there
is only a unique state compatible with the measurement results, which
turns out to be pure (or low rank).

There is also another clear physical meaning for the states that are
UDA by measuring A. Consider a Hamiltonian of the form

HA=m∑i=1αiAi.

(6)

Then any unique ground state |ψ⟩ of HA is UDA
by measuring A. This is easy to verify: if there is any
other state ρ that gives the same measurement results, then
ρ has the same energy as that of |ψ⟩, which is the
ground state energy. Therefore, any pure state in the range of ρ
must also be a ground state, which contradicts the fact that
|ψ⟩ is the unique ground state. In other words, UDA is a
necessary condition for |ψ⟩ to be a unique ground state of
HA. It is in general not sufficient, but the exceptions
are likely rare Chen et al. (2012, 2012a).

The uniqueness properties for pure states, for both UDP and UDA, have
also been studied extensively in the case of multipartite quantum
systems, where the observables correspond to reduced density matrices
(RDMs). That is, the observables are chosen to act nontrivially on
only some subsystems. For an n-particle system and a constant k<n,
there are a total of (nk)k-RDMs, and the corresponding
measurements A are those ≤k-body operators. For
example, for a three-qubit system and k=2, one can choose
A as all the one and two-particle Pauli operators. Of
course, one can also choose to look at some of the (nk)-RDMs,
rather than all of them. For instance, for a three-particle system,
one can look at 2-RDMs of particle pairs {1,2} and {1,3}.

It is known that almost all three-qubit pure states are UDA by their
2-RDMs Linden et al. (2002). These authors also show that UDP implies UDA
for three-qubit pure states, for 2-RDMs. This result can be further
improved to 2-RDMs of particle pairs {1,2} and
{1,3}Chen et al. (2012). More generally one can consider a
three-particle system of particles 1,2,3 with Hilbert spaces whose
dimensions are d1,d2,d3, respectively. If d1≥d2+d3−1,
then almost all pure states are UDA by their 2-RDMs of particle
pairs {1,2} and {1,3}. In contrast, if d1≥2, then
almost all pure states are UDP by their 2-RDMs of particle pairs
{1,2} and {1,3}, as shown by Diosi Diósi (2004).

For n-particle quantum systems with equal dimensional subsystems,
almost all pure states are UDA by their k-RDMs of just over half of
the parties (i.e., k∼n/2). Furthermore, ∼n/2 properly
chosen RDMs among all the (nk)k-RDMs suffice Jones and Linden (2005).
W-type states are UDA by their 2-RDMs, and n−1 of those 2-RDMs
are enough Parashar and Rana (2009). General symmetric Dicke states are UDA by
their 2-RDMs Chen et al. (2012b). It has been shown that the only
n-particle pure states which cannot be UDP by their (n−1)-RDMs are
those GHZ-type states, and the result is further improved to the case
of UDA Walck and Lyons (2008). Their results also show that UDP implies UDA for
n-qubit pure states, for (n−1)-RDMs.

Despite these many results, there is no systematic study of these two
different types of uniqueness for pure states. This will be the focus
of this paper, where we are interested in knowing for given
measurements A, whether UDP and UDA are the same, or are
different. We will give a general argument that there is a gap between
the number of observables needed for the two different cases. However,
in many interesting circumstances, they can coincide. Our discussions
extend naturally to the case of low rank quantum states instead of
just pure states. Here one can also look at two kinds of uniqueness
when measuring given observables A: one is uniqueness among
all low rank states, the other is among all states of any rank.

We organize the paper as follows. In Sec. II, we first show that there
is a set of 5d−7 observables that insures every pure state is UDA;
which should be compared to the UDP result 4d−5. Thus in general
there is a gap between the optimal results for the UDP and UDA cases,
and we illustrate this with some examples. Sec. III discusses the case
of observables corresponding to RDMs of a multipartite quantum state,
where for the three particle case, we show that if
d1≥min(d2,d3), then almost all pure states are UDA by their
2-RDMs of particle pairs {1,2} and {1,3}, improving the
bounds given in Linden and Wootters (2002). However this still leaves a gap with the
Diosi result for the case of UDP in Diósi (2004). We further discuss
circumstances where UDP can imply UDA for all pure states. In Sec. IV,
we show that when there are only two independent measurements
performed, then UDP always implies UDA, by making use of convexity of
the numerical range of operators. In a more general case, if there is
a compact group of symmetries of the state space which has the span of
the operators measured as its set of fixed points, then UDP implies
UDA for all pure states. We analyze the possible dimensions for those
fixed point sets. A summary and some discussions are included in Sec.
VI.

In this section, we discuss the minimum number of observables needed
to have all pure states be UDA. We start by choosing a Hermitian basis
{λi}d2−1i=0 for the operators on Hd.
Without loss of generality we choose λ0=√d−1I, the
identity operator on Hd, which has trace d. We further
require that the λi’s are orthogonal, in the sense that for
i,j≥0,

trλiλj=d(d−1)δij.

(7)

The d×d Hermitian matrices form a real inner product space
with inner product ⟨A,B⟩=tr(AB), so such a basis
{λi}d2−1i=0 exists for any dimension d. For
instance, for the qubit case (d=2), we can choose the Pauli basis

λ1=(0110),λ2=(0−ii0),λ3=(100−1).

(8)

For the qutrit case (d=3), one can choose λi=√3Mi
for i>0, where Mis are the Gell-Mann matrices given by

For general d, one can choose λi=√d(d−1)2Mi
for i>0, where Mis are the generalized Gell-Man matrices.

We can now write any density operator ρ as

ρ=1d(I+→r⋅→λ),

(10)

where →λ=(λ1,λ2,…,λd2−1),
and where →r=(r1,r2,…,rd2−1) has real entries.

We have trρ2≤1, therefore →r⋅→r≤1, and
the equality holds if ρ is a pure state. However, not every state
satisfying →r⋅→r=1 is a pure state. Indeed, ρ is
a pure state if and only if ρ2=ρ, which gives equations that
→r needs to satisfy.

If one of the observables is a multiple of the identity, then we can
drop it from the list of observables without affecting UDA and UDP. If
two states agree on an observable Ai, then they agree on Ai+tI for any real scalar t, so we can adjust each of the observables
A=(A1,…,Am) to have trace zero without affecting
UDA or UDP. Hence hereafter we assume all Ai are traceless.

For any observable Ai, we can expand in terms of {λi} as

Ai=d2−1∑j=1αijλj.

(11)

Then the average value of Ai is given by

tr(Aiρ)=1d(d+∑jrjαijd(d−1))=1+(d−1)→r⋅→αi,

(12)

where →αi={αi1,αi2,…,αi(d2−1)}.

To discuss the problem for any pure state to be UDA, the constant 1
and constant factor d−1 can be ignored, as these are the same
constants for all states. Therefore we have

tr(Aiρ)∼→r⋅→αi,

(13)

where ∼ means that the average value of Ai for the state
ρ is geometrically equivalent to the projection of →r onto
→αi.

Alternatively, define T:Rd2−1→Rm by
T(→r)=(→r⋅α1,…,→r⋅→αm). Let L be the linear subspace of Rd2−1
spanned by →α1,…,→αm, and let π be
the orthogonal projection from Rd2−1 onto L. Then
π and T have the same kernel, namely L⊥. Thus for states
ρ1,ρ2, we have T(ρ1)=T(ρ2) if and only if
π(ρ1)=π(ρ2), so in considering UDA and UDP we can
treat T as being the orthogonal projection onto L.

If we subtract the density matrix I/d from all states, then the
translated set of states sits in the real d2−1 dimensional subspace
of trace zero Hermitian matrices. In this sense, we are actually
working with real geometry in Rd2−1. All quantum states
then sit inside the d2−1-dimensional unit ball, with pure states
corresponding to unit vectors, but not every vector on the unit
d2−2-dimensional sphere is a pure state. The observables span an
m-dimensional subspace that all the quantum states will be projected
onto. We will simply say the subspace is spanned by A when
no confusion arises, and we will no longer distinguish an operator
Ai from the corresponding vector →αi. Indeed we only
consider the real span of A, and we denote it by
S(A). For each S(A), there
is an orthogonal subspace in Rd2−1 of dimension
d2−1−m, which we denote by S(A)⊥. Here
we are taking the orthogonal complement in the space of traceless
Hermitian matrices, so that every
V∈S(A)⊥ is traceless.

We now are ready to state our first theorem.

Theorem 1.

For a d-dimensional system (d>2), there exists a set of 5d−7
observables for which every pure state is UDA.

To see why this is the case, note that in the above-mentioned
geometrical picture, it is clear that a pure state
|ψ⟩⟨ψ| is UDA by measuring A if there does
not exist any operator V∈S(A)⊥, such
that |ψ⟩⟨ψ|+V is positive. One sufficient condition
will then be that any operator V∈S(A)⊥
has at least two positive and two negative eigenvalues. We will use
this sufficient condition to construct a desired
S(A)⊥.

In order to construct S(A)⊥, we provide a
set of m=d2−5d+6 linearly independent Hermitian matrices H1,H2,…,Hm∈Md(C) explicitly, such that the Hermitian
matrix

m∑j=1rjHj

has at least two positive eigenvalues for any nonzero real vector
r=(rj)∈Rm.

Our construction is motivated by and similar to the diagonal filling
technique used in Ref. Cubitt et al. (2008), but along the other direction of
the diagonals.

This then means that measuring d2−1−(d2−5d+6)=5d−7 observables is
enough for any pure state to be UDA, which proves the theorem. There
are indeed technical details to be clarified that we leave to Appendix
A.

If we compare our results with those given in Heinosaari et al. (2011), which
shows that measuring 4d−5 observables are enough for any pure state
to be UDP, there exists an obvious gap. We claim that this gap indeed
cannot be closed in general. To see this, let us look at the simplest
case of d=3, where the results just compared state that 7
observables are enough for any pure state to be UDP but 8
observables are enough for any pure state to be UDA.

If one can measure a particular set A with 7 observables
and have all pure states be UDA, then also every state also must be
UDP for measuring A. According to Heinosaari et al. (2011), this only
happens if S(A)⊥ contains a single
invertible traceless operator V, meaning V is rank 3. Without
loss of generality we can assume the largest eigenvalue V to be
positive with an eigenstate |ψ⟩. Then |ψ⟩ is not UDA
by measuring A since as observed in Heinosaari et al. (2011) there
exists a mixed state which also has the same average values as those
of |ψ⟩. Therefore, one cannot only measure 7 observables
for all pure states to be UDA.

For general d, our construction needs 5d−7 observables. We do not
know whether this is the optimal construction, but it is very unlikely
one can get this down to 4d−5. In other words, in general UDA and
UDP for pure states should be indeed two different concepts and there
should always be gaps between the number of observables needed to be
measured for each case to uniquely determine any pure quantum state.
This is one exception though, which is for the qubit case (i.e.,
d=2) where it is shown in Heinosaari et al. (2011) that for all pure states to
be UDP, one needs to measure 3=22−1 variables, which then uniquely
determine any quantum state among all states.

Finally, we remark that our results in Theorem 1 naturally extend to
the case of low rank states. That is, for a rank q<d/2 quantum state
ρ, we can similarly consider two different cases: (1) ρ is
uniquely determined by measuring A among all rank ≤q
states (which was considered in Heinosaari et al. (2011)) (2) ρ is uniquely
determined by measuring A among all quantum states of any
rank.

Theorem 2.

For a d-dimensional system (d>2) measuring (4q+1)d−(4q2+2q+1)
observables is enough for a rank ≤q state to be uniquely
determined among all states.

Compared to the results in Heinosaari et al. (2011), where 4q(d−q)−1 observables
are needed to uniquely determine any rank ≤q states among all
rank ≤q states, when d is large the difference in the leading
term has a d gap. The proof idea is similar to that of
Theorem 1, so we leave the details to Appendix A.

In this section we discuss the case where the Hilbert space
Hd is a multipartite quantum system, where the
observables correspond to the reduced density matrices (RDMs). That
is, the observables are chosen to be acting nontrivially only on some
subsystems. For instance, for a three-qubit system, the observables
corresponding to the 2-RDMs of particle pairs {1,2} can be
chosen as

A=

(X1,X2,Y1,Y2,Z1,Z2

(14)

X1X2,X1Y2,X1Z2,Y1X2,Y1Y2,

Y1Z2,Z1X2,Z1Y2,Z1Z2),

where Xi,Yi,Zi are Pauli X,Y,Z operators acting on the ith
qubit.

For simplicity in this section we consider only 3-particle systems,
labeled by 1,2,3, and each with Hilbert space dimension
d1,d2,d3, respectively. That is, Hd=Hd1⊗Hd2⊗Hd3
and d=d1d2d3. Nevertheless, our method naturally extends to
systems of more than 3-particles.

Recall that for a three particle system, it is known that almost all
three-qubit pure states are UDA by their 2-RDMs Linden et al. (2002). This
result can be further improved to 2-RDMs of particle pairs {1,2}
and {1,3}Chen et al. (2012). More generally, if d1≥d2+d3−1,
then almost all pure states are UDA by their 2-RDMs of particle
pairs {1,2} and {1,3}Linden and Wootters (2002). In contrast, if d1≥2, then almost every pure state is UDP by its 2-RDMs of particle
pairs {1,2} and {1,3}Diósi (2004).

We notice that different from the discussion in Sec. II, one no longer
considers uniqueness for all pure states, but ‘almost all’ of them.
This means there exists a measure zero set of pure states which are
not uniquely determined. For instance, for the three qubit case, any
state which is local unitarily equivalent to the GHZ type state

|GHZ⟩type=a|000⟩+b|111⟩

(15)

cannot be UDP, as any state of the form
a|000⟩+beiθ|111⟩ has the same 2-RDMs as those of
|GHZ⟩type. This means that, for a three qubit pure state
|ψ⟩, it is either UDA, or not UDP. In other words, if any
three qubit pure state |ψ⟩ is UDP, then it is UDA by its
2-RDMs of particle pairs {1,2} and {1,3}. In this sense, we
say in this case UDP implies UDA for all pure states.

However, for the general case of a three particle system, there is a
gap between known results of UDA and UDP. Our following result
improves the bound for the UDA case.

Theorem 3.

If d1≥min(d2,d3), then almost every tripartite quantum state
|ϕ⟩∈Hd1⊗Hd2⊗Hd3 is
UDA by its 2-RDMs of particle pairs {1,2} and {1,3},

To see why this is the case, an arbitrary pure state |ϕ⟩ of
this system can be written as

|ϕ⟩123=d1∑i=1d2∑j=1d3∑k=1cijk|i⟩1|j⟩2|k⟩3.

(16)

If there is another state ρ which agrees with |ϕ⟩ in its
subsystems {1,2} and {1,3}, then we can find a pure state
|ψ⟩1234∈Hd1⊗Hd2⊗Hd3⊗Hd4 which agrees with ρ
on the subsystem {1,2,3} and also agrees with |ϕ⟩123
in subsystems {1,2} and {1,3}.

Since the rank of the 2-RDM of the subsystem {1,2} is at most
d3, the pure state |ψ⟩1234 can be written as a
superposition of |vl⟩|El⟩ as follows.

|ψ⟩1234=d3∑l=1|vl⟩|El⟩

(17)

where

|vl⟩=d1∑i=1d2∑j=1cijl|i⟩1|j⟩2

(18)

for any 1≤l≤d3. Here {|El⟩}d3i=1 will
be vectors (perhaps unnormalized) in Hd3⊗Hd4.

The states {|El⟩}d3i=1 can be chosen to be
orthonormal vectors in the subsystem Hd3⊗Hd4, and then for almost all states |ϕ⟩, the
set of {|vl⟩}d3i=1 will be linearly independent. Let
us write |El⟩=d3∑k=1|k⟩3|elk⟩4. For any 1≤l≤d3, we will have

|ψ⟩1234=d1∑i=1d2∑j=1d3∑k,l=1cijl|i⟩1|j⟩2|k⟩3|elk⟩4.

(19)

Now let’s consider the subsystem {1,3}. Since |ϕ⟩123
and |ψ⟩1234 have the same RDMs for particles {1,3},
this gives

tr2|ϕ⟩⟨ϕ|=tr{2,4}|ψ⟩⟨ψ|.

(20)

Substituting Eqs. (16) and (17) into
Eq. (20), and comparing each matrix element, results in
the following equalities (for all m,m′,n,n′):

d2∑j=1cmjnc∗m′jn′=d2∑j=1d3∑k,k′=1cmjkc∗m′jk′⟨ek′n′|ekn⟩.

(21)

Now let us define xijkl=⟨eij|ekl⟩. Then
Eq. (21) is a linear equation system with variables
xijkl. It is not hard to verify that

xijkl={1if i=j,k=l0otherwise

(22)

is a solution to the equation system, which corresponds to the state
|ϕ⟩123.

Now we need to show that when d1≥min{d2,d3},
Eq. (21) has only one solution which is given by
Eq. (22). It turns out that this is indeed the case
which then proves Theorem 3. In fact, the linear equations
above are generically linearly independent. To see this, let’s fix n,n′ and m,m′, the right-hand side of
Eq. (21) is d3∑k,k′=1⟨α|m′k′⋅|α⟩mkxk′n′kn where
|α⟩mk=d2∑j=1cmjk|j⟩. Then the
coefficient matrix can be written as the following:

If there are more than 1 solutions, then the determinant of the
above matrix should be zero. Note that the determinant can be written
as a polynomial of cmjk’s and c∗m′jk′’s. Since ∏ciii appears only once in the polynomial, the determinant of the
top d23 by d23 submatrix must be non-zero generically.
Therefore, d21d23 linear equations are sufficient to determine
d43 variables.

However, we do not know whether the sufficient condition given by
Theorem 3 for almost all three-particle pure state to be
UDA by its 2-RDMs of particle pairs {1,2} and {1,3} is also
necessary. This still leaves a gap between the result of
Theorem 3 for UDA, and the result for UDP in Diósi (2004).
They both only coincide when d1=d2=d3=2, i.e., the three qubit
case. It remains open for other cases, whether UDP can imply UDA.

Following a similar discussion as in Sec. II, our result in this
section also extends to uniqueness of low rank quantum states. In
particular, we have the following theorem.

Theorem 4.

Almost every tripartite density operator ρ acting on the Hilbert
space Hd1⊗Hd2⊗Hd3 with rank no more than ⌊d1d3⌋ can be uniquely determined among all
states by its 2-RDMs of particle pairs {1,2} and {1,3}.

This result is to our knowledge, the first one for uniqueness of mixed
states with respect to RDMs. The proof is a direct extension of that
for Theorem 3, but with more lengthy details that we
will include in Appendix B.

Let us look at some consequences of Theorem 4. Consider a
four qubit system with qubits 1,2,3,4, and look at the qubits 3,4
as a single systems 3′. Then Theorem 4 says also that
almost all four qubit states of rank 2 are UDA by their RDMs of
particles {1,2} and {1,3′}={1,3,4}, or one can say that
almost all four qubit states of rank 2 are UDA by their 3-RDMs.
This is indeed consistent with the multipartite result in Jones and Linden (2005)
which states that almost all four-qubit pure states are UDA by their
3-RDMs, and our result is indeed stronger. This demonstrates
that our analysis naturally extends to systems of more than
3-particles. We also remark that the rank of a state ρ which
could be UDA by its k-RDMs needs to be relatively low, otherwise one
can always find another state ρ′ with lower rank which has the
same k-RDMs as those of ρChen et al. (2012c).

In Sec. II and Sec. III, we discussed the difference and coincidence
between the two kinds of uniqueness for pure states, UDA and UDP,
which in general are not the same thing. However, in certain
interesting circumstances such as the three qubit case with respect to
2-RDMs, and in general the n-qubit case respect to (n−1)-RDMs,
they do coincide. Starting from this section we would like to build
some general understanding of the circumstances when UDP implies UDA
for all pure states.

We start from the simplest case of m=2, where only two observables
are measured, i.e., A=(A1,A2). Intuitively, in this
extreme case almost no pure state can be uniquely determined, either
UDA or even UDP. However there are also exceptions. For instance, if
one of the observables, say A1, has a nondegenerate ground state
|ψ⟩, then |ψ⟩ is UDA (hence, of course, UDP) even by
measuring A1 only. One would hope this is the only exception, that
is, for a pure state |ψ⟩, either it is UDA, or it is not UDP,
when only two observables are measured. We make this intuition
rigorous by the following theorem.

Theorem 5.

When only two observables are measured, i.e., A=(A1,A2),
UDP implies UDA for any pure state |ψ⟩, regardless of the
dimension d.

To prove this theorem, recall that measuring A (i.e.,
measuring every observable in A) for all quantum states
ρ returns the set Cm(A) given by Eq. (3).
We know that Cm(A) is a convex set, meaning for any
→x,→y∈Cm(A), we have
(1−s)→x+s→y∈Cm(A) for any 0<s<1.

For pure states, the corresponding set of average values is given by
Wm(A) as defined in Eq. (5). Unlike
Cm(A), Wm(A) in general is not convex.
Nevertheless, it is easy to see that Wm(A)=Cm(A)
when Wm(A) is convex.

For m=2, the Hausdorff–Toeplitz theorem
Toeplitz (1918); Hausdorff (1919) gives convexity of the
numerical range of any operator, which in turn shows that
W2(A) is convex. We explain it briefly here. For any
operator B acting on a Hilbert space Hd, the numerical
range of B is the set of all complex numbers
⟨ψ|B|ψ⟩, where |ψ⟩ ranges over all pure
states in Hd.

Note that one can always write B as

B

=

12[(B+B†)+(B−B†)]

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=

12[(B+B†)+i(−iB+iB†)].

If we define A1:=(B+B†)/2 and A2:=(−iB+iB†)/2 then
clearly both A1 and A2 are Hermitian. Then W2(A) is
nothing but the numerical range of B=A1+iA2 and hence is
convex.

Furthermore, by studying the properties of the numerical range, it was
shown in Embry (1970) (using different terminology) that if a
pure state |ψ⟩ is UDP, the point
→x:=A(|ψ⟩) must be an extreme point of
W2(A). Here →x is an extreme point of the convex
set W2(A) if there do not exist →y,→z∈W2(A), such that →x=(1−s)→y+s→z for some
0<s<1.

Because W2(A)=C2(A), →x is also an
extreme point of C2(A). One can further show that for any
extreme point →x of C2(A), and any quantum state
ρ with A(ρ)=→x, any pure quantum state
|ϕ⟩ in the range of ρ will also have
A(|ϕ⟩)=→x. This then implies that if a pure
state |ψ⟩ is UDP by measuring A, it must also be
UDA, which proves the theorem.

Again, all the technical details of the proof will be presented in
Appendix C.

In an attempt to extend Theorem 5 to the m≥3 case, a
natural question that one could ask is whether or not UDP implies UDA
whenever Wm(A) is convex. Unfortunately this is not the
case, as demonstrated by the following example.

For the qutrit case (d=3), consider the observables
A=(M1,M2,M3), where the Mis are the Gell-Mann
matrices given in Eq. (9). These are the Pauli
operators embedded in the qutrit space. It is easily verified that in
this case, Wm(A) is the Bloch sphere together with its
interior and is thus convex. Nonetheless, the unique pure state
compatible with measurement result (0,0,0) is the state |2⟩,
even though there are many mixed states sharing this measurement
result, such as 12(|0⟩⟨0|+|1⟩⟨1|).

Therefore, although the Hausdorff–Toeplitz theorem
Toeplitz (1918); Hausdorff (1919) is famous for showing the
convexity of numerical range of any operator, there is indeed a deeper
reason than just the convexity of the numerical range which governs
the validity of Theorem 5. We leave the more detailed
discussion to Appendix C.

In this section, we discuss some circumstances where UDP implies UDA
in a more general context where more than two observables are
measured, i.e., m>2. Our focus is on the symmetry of the set of all
quantum states. For a d-dimensional Hilbert space Hd we
denote this set of states by Kd, that is

Kd={ρ∣ρactsonHd,tr(ρ)=1}.

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Note that Kd is convex, as we know that for any ρ1,ρ2∈Kd, (1−s)ρ1+sρ2∈Kd for all 0<s<1. Furthermore, the
extreme points of Kd are all the pure states. Kd is also called
the state space for all the operators acting on Hd.

We now explain the intuition. If Kd has a certain symmetry, then
two pure states |ψ1⟩ and |ψ2⟩ that are ‘connected’
by the symmetry will give the same measurement results, and states
|ψ⟩ fixed by the symmetry will also be fixed by the
projection onto the space of observables. In this situation, UDP
implies UDA for all pure states.

To make this intuition concrete, let us first consider an example for
d=2, i.e., the qubit case. We know that Kd can be parameterized
as in Eq. (10), where for d=2,
λ1=X,λ2=Y,λ3=Z are chosen as Pauli matrices
given in Eq. (8). Here Kd is the Bloch ball as shown
in FIG. 1. The Bloch ball is clearly a convex set and
the extreme points are those pure states on the boundary, which give
the Bloch sphere.

Figure 1: Symmetry of the Bloch ball

We know that geometrically, measuring the observables in A
corresponds to the projection onto the plane spanned by A.
For example, if we measure the Pauli X and Y operators, then
geometrically this corresponds to the projection of the Bloch ball
onto the xy plane. Since the Bloch ball has reflection symmetry with
respect to the xy plane, two pure states (e.g. points B and C)
connected by that symmetry will project onto the same measurement
result P, as will all mixtures of B and C. Hence neither UDP nor
UDA hold for such pure states for measuring X and Y. On the other
hand, pure states fixed by the reflection symmetry are also fixed by
the projection onto the xy plane. These are precisely the points on
the Bloch sphere that are in the xy plane (e.g. the points E and
G in FIG. 1), and for such pure states both UDP and
UDA hold. Therefore, for the observables X,Y we conclude that UDP =
UDA.

Now let us look at another case where we only measure the Pauli X
operator. Consider the group of symmetries of the Bloch ball
consisting of rotation around the x axis. (Rotation by angle
α, is shown in FIG. 1. In that figure, point
B will become point F after this particular rotation, and indeed
both points B and F yield the same measurement result, which is
represented by point P on the x axis.) Note that two points on the
Bloch sphere will project to the same measurement result on the x
axis if and only if they are in the same orbit under the rotation
group. Thus a measurement result will come from a single pure state
exactly when that pure state is a fixed point, and hence either both
or neither of UDP and UDA hold for each pure state. For example, the
point E is fixed by the rotation, and is uniquely determined by the
measurement of X among all states. E corresponds to the −1
eigenstate of the Pauli X operator. Therefore, the rotational
symmetry of the Bloch ball along the x axis gives UDP =UDA for any
pure state when measuring the Pauli X operator, which corresponds to
the x axis.

Mathematically, a symmetry of Kd is an affine automorphism of
Kd. If U∈Md is unitary, the map taking ρ to UρU† is such an affine automorphism (which for d=2 will just
be rotation around some axis of the Bloch ball). For instance, the
rotation symmetry along the x axis by an angle α is given by
conjugation by the unitary operator exp(−iXα/2). If V is the
conjugate linear map given by complex conjugation in the computational
basis (V|ψ⟩=|ψ∗⟩), then the map taking ρ to
VρV† is the transpose map. For d=2, this map is
reflection of the Bloch ball in the xy-plane.

Recall that for a set of observables A, we denote the real
linear span by S(A). When discussing the
uniqueness problems, it makes no difference if we append the identity
operator to A. Let us then assume A=(I,A1,…,Am). We are now ready to put our intuition into a
theorem.

Theorem 6.

Assume there exists a compact group G of affine automorphisms of
Kd whose fixed point set is Kd⋂S(A). Then each
pure state acting on Hd which is UDP for measuring
A is also UDA.

In the first example above, the group for the reflection consists of
the two element group generated by the reflection. In the rotation
example, we can take the group to consist of all rotations around the
given axis. We will leave the detailed mathematical proof of Theorem
6 to Appendix D, where operator algebras are one
ingredient of the proof.

To motivate some further consequences of Theorem 6,
consider a simple example. If A consists of a basis of
diagonal matrices (i.e., a set of mutually commuting observables),
then for any pure state, UDP implies UDA by Theorem 6.
Here the group of symmetries can be taken to be conjugation by all
diagonal unitaries. This group has fixed point set Kd⋂S(A). In a more general case, if the complex span of S(A) is a *-subalgebra of the operators acting on
Hd, then UDP = UDA for all pure states for measuring
A. This is a natural corollary of Theorem 6
that we will also discuss in detail in Appendix D.

In this work, we have discussed the uniqueness of quantum states
compatible with given results for measuring a set of observables. For
a given pure state, we consider two different types of uniqueness, UDP
and UDA. We have taken the first step to study their relationship
systematically. In doing so we have established a number of results,
but also leave with many open questions.

First of all, although in general UDP and UDA are evidently different
concepts, their difference is surprisingly ‘not that large’.
Specifically in the sense of general counting of the number of
variables one needs to measure to uniquely determine all pure states
in a d dimensional Hilbert space. Compared to full quantum
tomography which requires d2−1 variables measured to uniquely
determine any quantum state, the 5d−7 observables we have
constructed to uniquely determine any pure state among all states is a
significant improvement. It is indeed larger than the 4d−5
observables given in Heinosaari et al. (2011) to uniquely determine any pure state
among all pure states, but the difference is only linear in d. We do
not know whether there could be another construction for which we
could further close the linear difference between UDA and UDP, to
leave only a constant gap for large d.

When the Hilbert space is a multipartite quantum system, and the
observables correspond to the RDMs, we focused on the situation when
‘almost all pure states’ are uniquely determined. We considered a
3-particle system with Hilbert space
Hd=Hd1⊗Hd2⊗Hd3,
and showed that if d1≥min(d2,d3), then almost all pure states
are UDA by their 2-RDMs of particle pairs {1,2} and {1,3}.
This improves the results of Linden and Wootters (2002), where d1≥d2+d3−1 is
required; however it still leaves a gap compared to the Diosi UDP
result which states that for d1≥2, almost all pure states are
UDP by their 2-RDMs of particle pairs {1,2} and {1,3}.
Because our proof only gives a sufficient condition for UDA, we do not
know whether it can be further improved. We also do not have an
example showing there is indeed gap between UDA and UDP for almost all
three-particle pure states to be uniquely determined by 2-RDMs of
particle pairs {1,2} and {1,3}.

Finally, we considered situations for which we can show that UDP
implies UDA. These include: (i) the general 2-qubit system; (ii) the
3-qubit system when we consider uniqueness for almost all pure
states and the measurements corresponds to 2-RDMs; (iii) when only
two observables are measured; and (iv) the observables measured
correspond to some symmetry of the state space. However we do not know
how far we are from enumerating all the possible situations that UDP
implies UDA, when considering uniqueness for all pure states or almost
all pure states. In principle one can even consider the relationship
between UDP and UDA for special subsets of pure states.

We believe our systematic study of the uniqueness of quantum states
compatible with given measurement results shed light on several
aspects of quantum information theory and its connection to different
topics in mathematics. These include quantum tomography and the space
of Hermitian operators, unique ground states of local Hamiltonians and
general solutions to certain linear equations, measurements and
numerical ranges of operators, and the geometric meaning of
measurements and the symmetry of state space. We thus conclude with
several open questions that we believe warrant further investigation.

Lemma 1.

There exists a set of m=d2−5d+6 linearly independent Hermitian matrices
H1,H2,…,Hm∈Md(C), such that the Hermitian matrix

m∑j=1rjHj

has at least two positive eigenvalues for any nonzero real vector
r=(rj)∈Rm.

Proof.

We prove the statement by giving an explicit construction. Our proof
is motivated by and similar to the diagonal filling technique used
in Ref. Cubitt et al. (2008), but along the other direction of the diagonals.

We will need the lemma 9 from Ref. Cubitt et al. (2008) about totally
non-singular matrix, which we restate as Lemma 3 in
the following. For simplicity, we also assume that the totally
non-singular matrix is real. Therefore, for any length L∈N
and L≥2, there is L−1 linearly independent real vectors such
that every nonzero linear combination of them has at least 2
nonzero entries.

Let H=(Hjj′)∈Md(C) be a matrix. We will always fix the
diagonal to be zero, namely Hjj=0 for 0≤j≤d−1. In the
upper triangular part of the matrix not including the diagonal,
there are 2d−3 lines of entries parallel to the antidiagonal. That
is, each line contains entries Hjj′ with j<j′ and j+j′=k
where k goes from 1 to 2d−3. We will call it the k-th line
of the matrix in the following. We also call the set of entries
Hjj′ with j+j′=k the k-th antidiagonal. It is easy to see
that the length Lk of the k-th line is

Lk=⎧⎪
⎪⎨⎪
⎪⎩[k+12]for k≤d−1,[2d−1−k2]otherwise.

So the length Lk≥2 for 3≤k≤2d−5, and we can find
Lk−1 real vectors for which every nonzero linear combination has
at least 2 nonzero entries. For each of the Lk−1 vectors, we
can form two Hermitian matrices. One of them is the symmetric one
whose k-th line is filled with the vector, and the lower
triangular part determined by the Hermitian condition. Such a matrix
is a real symmetric matrix having nonzero entries only on the k-th
antidiagonal. We will call it a real k-th line matrix. The other
is the one with k-th line filled with the vector multiplied by
i=√−1, and lower part is determined by the Hermitian
condition. This is a matrix consisting of purely imaginary entries
on the k-th antidiagonal and we call it an imaginary k-th line
matrix.

Now we prove that the constructed matrices satisfy our requirement.
First we prove that the matrices are linearly independent. It
suffices to show that the matrices of nonzero k-th line is
linearly independent. Let {vj} be the set of linearly
independent real vectors chosen for the k-th line. We need to show
that {(vj,vj),(ivj,−ivj)} is linearly independent
over C. If the contrary is true, that is, there exists complex
numbers cj,dj not all zero such that

∑jcj(vj,vj)+∑jdj(ivj,−ivj)=0.

This is equivalent to

∑jcjvj+i∑jdjvj=0∑jcjvj−i∑jdjvj=0.

From the above two equations, we get ∑jcjvj=0 and
∑jdjvj=0 which is a contradiction.

Next, we prove that for any nonzero real coefficient r∈Rm,
the matrix H=∑rjHj has at least two positive eigenvalues.
Let k0 be the largest k such that there is a k-th line matrix
Hj whose coefficient rj is nonzero. Then, either the real
k0-th line matrices or the imaginary ones have nonzero
coefficients. By the construction, this implies that there is at
least two nonzero entries on the k0th line of the matrix H. Let
the nonzero entries be a,b∈C. We then have a principle
submatrix of H that has the form

⎛⎜
⎜
⎜
⎜⎝0xya¯x0b0¯y¯b00¯a000⎞⎟
⎟
⎟
⎟⎠,

where x,y are two unknown number and ¯a represents the
complex conjugate of a. This matrix has trace 0 and determinant
|ab|2. Therefore, it has exactly two positive eigenvalues. As it
is a principle submatrix of matrix H, follows from
Theorem 7, H has at least two positive eigenvalues.

The number of matrices thus constructed is the summation

m=2d−5∑k=32(Lk−1),

which can be computed to be

d2−5d+6.

∎

Discussion: We note that our construction will also imply that the
matrix has at least two negative eigenvalues, thus at least rank 4.
But our bound is even better than the (d−3)2 bound on the dimension
of subspaces in which every matrix has rank ≥4. This is not a
contradiction as we are considering all real combinations. For
example, the case of d=4 has two matrices for our purpose, namely

These two matrices do satisfy our requirements, but their span
over C contains a rank 2 matrix H1+iH2.

Generalization: Similarly, length Lk≥q+1 for 2q+1≤k≤2d−2q−3, and we can find Lk−q real vectors for which every nonzero
linear combination has at least q+1 nonzero entries. For each of the
Lk−q vectors, we can also form two Hermitian matrices. Such
constructed matrices are linearly independent and any real linear
combination has at least q+1 positive eigenvalues.

Lemma 2.

There exists a set of m=d2−(4q+1)d+(4q2+2q) linearly
independent Hermitian matrices H1,H2,…,Hm∈Md(C),
such that the Hermitian matrix

m∑j=1rjHj

has at least q+1 positive eigenvalues for any nonzero real vector
r=(rj)∈Rm.

We just follow the lines of the proof of Lemma 1. To
complete our argument, we need to show that any 2(q+1) by 2(q+1)
invertible traceless, Hermitian, upper left triangler matrix has
exactly q+1 positive eigenvalues.

Let’s prove this claim by induction. When q=1, it is already known.
Let’s assume this claim holds true for any q≤r. Then for
q=r+1, we can write such matrix A in the following form